Let n greater than or equal to 1 be an integer and let G be a graph of
order p. A set D of vertices of G is a total n-dominating set of G if
every vertex of V(G) is within distance n from some vertex of D other
than itself. The minimum cardinality among all total n-dominating set
s of G is called the total n-domination number and is denoted by gamma
(n)(t)(G). A set S of vertices of G is n-independent if the distance (
in G) between every pair of distinct vertices of S is at least n + 1.
The minimum cardinality among all maximal n-independent sets of G is c
alled the it-independence number of G and is denoted by in(G). In this
paper, we present an algorithm for finding a total n-dominating set D
and a maximal n-independent set S in a connected graph with at least
p greater than or equal to 2n + 1 vertices. It is shown that these set
s D and S satisfy the inequality /S/ + n/D/ less than or equal to p. U
sing this result, we conclude that if G is a connected graph on p grea
ter than or equal to 2n + 1 vertices, then i(n)(G) + n . gamma(n)(t)(G
) less than or equal to p.